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<H1><A NAME="SECTION03480000000000000000"></A><A NAME="secnonsym"></A>
<BR>
Error Bounds for the Nonsymmetric Eigenproblem 
</H1>

<P>
The nonsymmetric eigenvalue
problem<A NAME="11350"></A> is more
complicated than the
symmetric eigenvalue problem. In this subsection,
we state the simplest bounds and leave the more complicated ones to
subsequent subsections.

<P>
Let <B><I>A</I></B> be an <B><I>n</I></B>-by-<B><I>n</I></B> nonsymmetric matrix, with eigenvalues

<!-- MATH
 $\lambda_1, \ldots , \lambda_n$
 -->
<IMG
 WIDTH="80" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img562.gif"
 ALT="$\lambda_1, \ldots , \lambda_n$">.
Let <B><I>v</I><SUB><I>i</I></SUB></B> be a right eigenvector
corresponding to <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.gif"
 ALT="$\lambda_i$">:

<!-- MATH
 $A v_i = \lambda_i v_i$
 -->
<IMG
 WIDTH="84" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img563.gif"
 ALT="$A v_i = \lambda_i v_i$">.
Let 
<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.gif"
 ALT="$\hat{\lambda}_i$">
and <IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img564.gif"
 ALT="$\hat{v}_i$">
be the corresponding
computed eigenvalues and eigenvectors, computed by expert driver routine
xGEEVX (see subsection&nbsp;<A HREF="node29.html#subsecdriveeig">2.3.4</A>).
<A NAME="11354"></A><A NAME="11355"></A><A NAME="11356"></A><A NAME="11357"></A>

<P>
The approximate error bounds<A NAME="footfnm 0"><SUP>4.10</SUP></A>for the computed eigenvalues are
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
| \hat{\lambda}_i - \lambda_i | \leq {\tt EERRBD}(i) \; \; .
\end{displaymath}
 -->


<IMG
 WIDTH="173" HEIGHT="31" BORDER="0"
 SRC="img565.gif"
 ALT="\begin{displaymath}
\vert \hat{\lambda}_i - \lambda_i \vert \leq {\tt EERRBD}(i) \; \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The approximate error
bounds<A NAME="11361"></A><A NAME="11362"></A>
for the computed eigenvectors <IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img564.gif"
 ALT="$\hat{v}_i$">,
which bound the acute angles between the computed eigenvectors and true
eigenvectors <B><I>v</I><SUB><I>i</I></SUB></B>, are
<A NAME="11364"></A>
<A NAME="11365"></A>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\theta ( \hat{v}_i , v_i ) \leq {\tt VERRBD}(i) \; .
\end{displaymath}
 -->


<IMG
 WIDTH="164" HEIGHT="31" BORDER="0"
 SRC="img566.gif"
 ALT="\begin{displaymath}
\theta ( \hat{v}_i , v_i ) \leq {\tt VERRBD}(i) \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
These bounds can be computed by the following code fragment:

<P>
<PRE>
      EPSMCH = SLAMCH( 'E' )
*     Compute the eigenvalues and eigenvectors of A
*     WR contains the real parts of the eigenvalues
*     WI contains the real parts of the eigenvalues
*     VL contains the left eigenvectors
*     VR contains the right eigenvectors
      CALL SGEEVX( 'P', 'V', 'V', 'B', N, A, LDA, WR, WI,
     $             VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
     $             RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
      IF( INFO.GT.0 ) THEN
         PRINT *,'SGEEVX did not converge'
      ELSE IF ( N.GT.0 ) THEN
         DO 10 I = 1, N
            EERRBD(I) = EPSMCH*ABNRM/RCONDE(I)
            VERRBD(I) = EPSMCH*ABNRM/RCONDV(I)
10       CONTINUE
      ENDIF
</PRE>

<P>
For example<A NAME="footfnm 0"><SUP>4.11</SUP></A>, if

<!-- MATH
 ${\tt SLAMCH('E')} = 2^{-24} = 5.961 \cdot 10^{-8}$
 -->
<IMG
 WIDTH="259" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img397.gif"
 ALT="${\tt SLAMCH('E')} = 2^{-24} = 5.961 \cdot 10^{-8}$">
and
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
A = \left( \begin{array}{ccc} 58 & 9 & 2 \\186 & 383 & 96 \\-912 & -1551 & -388 \end{array} \right)
\end{displaymath}
 -->


<IMG
 WIDTH="235" HEIGHT="73" BORDER="0"
 SRC="img567.gif"
 ALT="\begin{displaymath}
A = \left( \begin{array}{ccc} 58 &amp; 9 &amp; 2 \\ 186 &amp; 383 &amp; 96 \\ -912 &amp; -1551 &amp; -388 \end{array} \right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
then true eigenvalues, approximate eigenvalues, approximate error bounds,
and true errors are
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER"><B><I>i</I></B></TD>
<TD ALIGN="CENTER"><IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.gif"
 ALT="$\lambda_i$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.gif"
 ALT="$\hat{\lambda}_i$"></TD>
<TD ALIGN="CENTER"><TT> EERRBD</TT><B>(<I>i</I>)</B></TD>
<TD ALIGN="CENTER">true 
<!-- MATH
 $| \hat{\lambda}_i - \lambda_i |$
 -->
<IMG
 WIDTH="67" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img531.gif"
 ALT="$\vert \hat{\lambda}_i - \lambda_i \vert$"></TD>
<TD ALIGN="CENTER"><TT> VERRBD</TT><B>(<I>i</I>)</B></TD>
<TD ALIGN="CENTER">true 
<!-- MATH
 $\theta ( \hat{v}_i , v_i )$
 -->
<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img568.gif"
 ALT="$\theta ( \hat{v}_i , v_i )$"></TD>
</TR>
<TR><TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER"><B>50</B></TD>
<TD ALIGN="CENTER"><B>50.00</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $8.5 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img569.gif"
 ALT="$8.5 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $2.7 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img570.gif"
 ALT="$2.7 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $2.6 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img571.gif"
 ALT="$2.6 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $8.5 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img572.gif"
 ALT="$8.5 \cdot 10^{-6}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">2</TD>
<TD ALIGN="CENTER"><B>2</B></TD>
<TD ALIGN="CENTER"><B>1.899</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $3.1 \cdot 10^{-1}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img573.gif"
 ALT="$3.1 \cdot 10^{-1}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.0 \cdot 10^{-1}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img574.gif"
 ALT="$1.0 \cdot 10^{-1}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.9 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img575.gif"
 ALT="$1.9 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $7.7 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img576.gif"
 ALT="$7.7 \cdot 10^{-5}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">3</TD>
<TD ALIGN="CENTER"><B>1</B></TD>
<TD ALIGN="CENTER"><B>1.101</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $3.1 \cdot 10^{-1}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img573.gif"
 ALT="$3.1 \cdot 10^{-1}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.0 \cdot 10^{-1}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img574.gif"
 ALT="$1.0 \cdot 10^{-1}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.8 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img577.gif"
 ALT="$1.8 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $7.5 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img578.gif"
 ALT="$7.5 \cdot 10^{-5}$"></TD>
</TR>
</TABLE>
</DIV>

<P>
<BR><HR>
<!--Table of Child-Links-->
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<UL>
<LI><A NAME="tex2html5466"
 HREF="node92.html">Further Details:  Error Bounds for the Nonsymmetric Eigenproblem</A>
<UL>
<LI><A NAME="tex2html5467"
 HREF="node93.html">Overview</A>
<LI><A NAME="tex2html5468"
 HREF="node94.html">Balancing and Conditioning</A>
<LI><A NAME="tex2html5469"
 HREF="node95.html">Computing <B><I>s</I></B> and <IMG
 WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img2.gif"
 ALT="${\rm sep}$"></A>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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